# Greatest lower bound

prove: "A nonempty set of real numbers bounded from below has a greatest lower bound."

cristo
Staff Emeritus
This looks like homework. What thoughts do you have on the question? What is the definition of the greatest lower bound of a set?

A lower bound of a non-empty subset A of R is an element d in R with d <= a for all a A.
An element m in R is a greatest lower bound or infimum of A if
m is a lower bound of A and if d is an upper bound of A then m >= d.

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cristo
Staff Emeritus
Ok, so here A is bounded below, so this tells you that there exists a lower bound to A. It may now be helpful to consider the set
-A:={-x:x∈A}, and use the completeness axiom to find the least upper bound on -A. The relationship between A and -A should help you find the greatest lower bound of A.

this is not number theory

this is not number theory, but it is instead mathematical analysis.

it is a very basic result that has been used to construct the real numbers and i think you will find it in any standard intro to analysis textbook (i personally recommend principles of mathematical analysis by Walter Rudin).

hope it helps

HallsofIvy